Non square linear system with two unknowns
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0
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I want to know under what conditions exists at least one solution for the following system
$$Ax + By = C$$
where $A$ is $p times q$ matrix, $B$ is $p times q$ matrix and $C$ is a vector of size $p$. Uniqueness doesn't matter. Thanks.
real-analysis linear-algebra systems-of-equations
edited Nov 14 at 15:48
Moo
5,2633920
asked Nov 14 at 14:58
Gustave
683211
add a comment |
up vote
0
down vote
favorite
I want to know under what conditions exists at least one solution for the following system
$$Ax + By = C$$
where $A$ is $p times q$ matrix, $B$ is $p times q$ matrix and $C$ is a vector of size $p$. Uniqueness doesn't matter. Thanks.
real-analysis linear-algebra systems-of-equations
edited Nov 14 at 15:48
Moo
5,2633920
asked Nov 14 at 14:58
Gustave
683211
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
I want to know under what conditions exists at least one solution for the following system
$$Ax + By = C$$
where $A$ is $p times q$ matrix, $B$ is $p times q$ matrix and $C$ is a vector of size $p$. Uniqueness doesn't matter. Thanks.
real-analysis linear-algebra systems-of-equations
edited Nov 14 at 15:48
Moo
5,2633920
asked Nov 14 at 14:58
Gustave
683211
I want to know under what conditions exists at least one solution for the following system
$$Ax + By = C$$
where $A$ is $p times q$ matrix, $B$ is $p times q$ matrix and $C$ is a vector of size $p$. Uniqueness doesn't matter. Thanks.
real-analysis linear-algebra systems-of-equations
real-analysis linear-algebra systems-of-equations
edited Nov 14 at 15:48
Moo
5,2633920
asked Nov 14 at 14:58
Gustave
683211
edited Nov 14 at 15:48
Moo
5,2633920
asked Nov 14 at 14:58
Gustave
683211
edited Nov 14 at 15:48
Moo
5,2633920
edited Nov 14 at 15:48
Moo
5,2633920
edited Nov 14 at 15:48
Moo
5,2633920
5,2633920
asked Nov 14 at 14:58
Gustave
683211
asked Nov 14 at 14:58
Gustave
683211
asked Nov 14 at 14:58
Gustave
683211
683211
add a comment |
add a comment |
1 Answer
1
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oldest
votes
up vote
2
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The system has a solution if and only if the system
$$begin{bmatrix}A& Bend{bmatrix}z=C$$ has a solution. Of course, the solution vector $z$ will have $2q$ entries, and if written as
$$z=begin{bmatrix}x\yend{bmatrix}$$
then $z$ is a solution of the second system if and only if $x, y$ are solutions to the original system.
You can interpret this result by looking at ${Ax|xinmathbb R^p}$ as "the set of all linear combinations of columns of $A$". Then, $By$ is the general element of the set of all linear combinations of columns from $B$, which means that $Ax+By$ can be any linear combination of columns from both $A$ and $B$.
answered Nov 14 at 15:01
5xum
88.4k392160
Is there anay rank condition on $A$, $B$ and $C$ to ensure the existence? thanks.
– Gustave
Nov 14 at 15:04
@Gustave Once you have the system $begin{bmatrix} A&Bend{bmatrix}z=C$, you can use the standard theorems from linear algebra to determine when a solution exists. Also, "rank condition" on $C$ makes little sense, since $C$ is a vector, not a matrix.
– 5xum
Nov 14 at 15:06
Thank you a lot sir.
– Gustave
Nov 14 at 15:07
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
2
down vote
The system has a solution if and only if the system
$$begin{bmatrix}A& Bend{bmatrix}z=C$$ has a solution. Of course, the solution vector $z$ will have $2q$ entries, and if written as
$$z=begin{bmatrix}x\yend{bmatrix}$$
then $z$ is a solution of the second system if and only if $x, y$ are solutions to the original system.
You can interpret this result by looking at ${Ax|xinmathbb R^p}$ as "the set of all linear combinations of columns of $A$". Then, $By$ is the general element of the set of all linear combinations of columns from $B$, which means that $Ax+By$ can be any linear combination of columns from both $A$ and $B$.
answered Nov 14 at 15:01
5xum
88.4k392160
Is there anay rank condition on $A$, $B$ and $C$ to ensure the existence? thanks.
– Gustave
Nov 14 at 15:04
@Gustave Once you have the system $begin{bmatrix} A&Bend{bmatrix}z=C$, you can use the standard theorems from linear algebra to determine when a solution exists. Also, "rank condition" on $C$ makes little sense, since $C$ is a vector, not a matrix.
– 5xum
Nov 14 at 15:06
Thank you a lot sir.
– Gustave
Nov 14 at 15:07
add a comment |
up vote
2
down vote
The system has a solution if and only if the system
$$begin{bmatrix}A& Bend{bmatrix}z=C$$ has a solution. Of course, the solution vector $z$ will have $2q$ entries, and if written as
$$z=begin{bmatrix}x\yend{bmatrix}$$
then $z$ is a solution of the second system if and only if $x, y$ are solutions to the original system.
You can interpret this result by looking at ${Ax|xinmathbb R^p}$ as "the set of all linear combinations of columns of $A$". Then, $By$ is the general element of the set of all linear combinations of columns from $B$, which means that $Ax+By$ can be any linear combination of columns from both $A$ and $B$.
answered Nov 14 at 15:01
5xum
88.4k392160
Is there anay rank condition on $A$, $B$ and $C$ to ensure the existence? thanks.
– Gustave
Nov 14 at 15:04
@Gustave Once you have the system $begin{bmatrix} A&Bend{bmatrix}z=C$, you can use the standard theorems from linear algebra to determine when a solution exists. Also, "rank condition" on $C$ makes little sense, since $C$ is a vector, not a matrix.
– 5xum
Nov 14 at 15:06
Thank you a lot sir.
– Gustave
Nov 14 at 15:07
add a comment |
up vote
2
down vote
up vote
2
down vote
The system has a solution if and only if the system
$$begin{bmatrix}A& Bend{bmatrix}z=C$$ has a solution. Of course, the solution vector $z$ will have $2q$ entries, and if written as
$$z=begin{bmatrix}x\yend{bmatrix}$$
then $z$ is a solution of the second system if and only if $x, y$ are solutions to the original system.
You can interpret this result by looking at ${Ax|xinmathbb R^p}$ as "the set of all linear combinations of columns of $A$". Then, $By$ is the general element of the set of all linear combinations of columns from $B$, which means that $Ax+By$ can be any linear combination of columns from both $A$ and $B$.
answered Nov 14 at 15:01
5xum
88.4k392160
The system has a solution if and only if the system
$$begin{bmatrix}A& Bend{bmatrix}z=C$$ has a solution. Of course, the solution vector $z$ will have $2q$ entries, and if written as
$$z=begin{bmatrix}x\yend{bmatrix}$$
then $z$ is a solution of the second system if and only if $x, y$ are solutions to the original system.
You can interpret this result by looking at ${Ax|xinmathbb R^p}$ as "the set of all linear combinations of columns of $A$". Then, $By$ is the general element of the set of all linear combinations of columns from $B$, which means that $Ax+By$ can be any linear combination of columns from both $A$ and $B$.
answered Nov 14 at 15:01
5xum
88.4k392160
edited Nov 14 at 15:04
answered Nov 14 at 15:01
5xum
88.4k392160
answered Nov 14 at 15:01
5xum
88.4k392160
answered Nov 14 at 15:01
5xum
88.4k392160
88.4k392160
Is there anay rank condition on $A$, $B$ and $C$ to ensure the existence? thanks.
– Gustave
Nov 14 at 15:04
@Gustave Once you have the system $begin{bmatrix} A&Bend{bmatrix}z=C$, you can use the standard theorems from linear algebra to determine when a solution exists. Also, "rank condition" on $C$ makes little sense, since $C$ is a vector, not a matrix.
– 5xum
Nov 14 at 15:06
Thank you a lot sir.
– Gustave
Nov 14 at 15:07
add a comment |
Is there anay rank condition on $A$, $B$ and $C$ to ensure the existence? thanks.
– Gustave
Nov 14 at 15:04
@Gustave Once you have the system $begin{bmatrix} A&Bend{bmatrix}z=C$, you can use the standard theorems from linear algebra to determine when a solution exists. Also, "rank condition" on $C$ makes little sense, since $C$ is a vector, not a matrix.
– 5xum
Nov 14 at 15:06
Thank you a lot sir.
– Gustave
Nov 14 at 15:07
Is there anay rank condition on $A$, $B$ and $C$ to ensure the existence? thanks.
– Gustave
Nov 14 at 15:04
Is there anay rank condition on $A$, $B$ and $C$ to ensure the existence? thanks.
– Gustave
Nov 14 at 15:04
@Gustave Once you have the system $begin{bmatrix} A&Bend{bmatrix}z=C$, you can use the standard theorems from linear algebra to determine when a solution exists. Also, "rank condition" on $C$ makes little sense, since $C$ is a vector, not a matrix.
– 5xum
Nov 14 at 15:06
@Gustave Once you have the system $begin{bmatrix} A&Bend{bmatrix}z=C$, you can use the standard theorems from linear algebra to determine when a solution exists. Also, "rank condition" on $C$ makes little sense, since $C$ is a vector, not a matrix.
– 5xum
Nov 14 at 15:06
Thank you a lot sir.
– Gustave
Nov 14 at 15:07
Thank you a lot sir.
– Gustave
Nov 14 at 15:07
add a comment |
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